A206927 Minimal numbers of binary length n+1 such that the number of contiguous palindromic bit patterns in the binary representation is minimal.

2, 4, 9, 18, 37, 75, 150, 300, 601, 1202, 2405, 4811, 9622, 19244, 38489, 76978, 153957, 307915, 615830, 1231660, 2463321, 4926642, 9853285, 19706571, 39413142, 78826284, 157652569, 315305138, 630610277, 1261220555

Offset: 1

Author

Hieronymus Fischer, Mar 24 2012

Keywords

• nonn
• base

Comments

Subsequence of A206926.

From left to right, the binary representation of a(n) consists of a concatenation of the bit pattern 100101 (=37). If the number of places is not a multiple of 6, the least significant places are truncated. This leads to a simple linear recurrence.

Example: a(19)=615830=10010110010110_2=concatenate('100101','100101','10')

Examples

			a(3)=9=1001_2 has 6 [=A206925(9)] contiguous palindromic bit patterns. This is the minimum value for binary numbers with 4 places and 9 is the least number with this property.
a(9)=601=1001011001_2 has 18 [=A206925(601)] contiguous palindromic bit patterns. This is the minimum value for binary numbers with 10 places and 601 is the least number with this property.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,1,-2).

Crossrefs

Cf. A006995, A206923, A206924, A206925, A206926, A070939.

Formula

a(n) = 37*2^(1+n mod 6)*(2^(6*floor(n/6))-1)/63 + floor(37*2^(n mod 6)/2^5).

a(n) = floor((37*2^(n+1)/63)) mod 2^(n+1).

A206925(a(n)) = 2*floor(log_2(a(n))).

a(n+1) = 2a(n) + floor(37*2^(n+2)/63) mod 2.

G.f.: x*( 2+x^2+x^4+x^5-2*x^6 ) / ( (x-1)*(2*x-1)*(1+x)*(x^2-x+1)*(1+x+x^2) ). - R. J. Mathar, Apr 02 2012

Also, g.f. x*(2+x^2+x^4+x^5-2*x^6)/((1-2*x)*(1-x^6)).

Extensions

Further formulas added by Hieronymus Fischer, Jan 13 2013